Foldy-Lax models are asymptotic models of wave scattering
by multiple obstacles, in the regime when their characteristic size tends to
zero. While frequency-domain Foldy-Lax models are now fairly well-studied
(see e.g. [1,2,3]), their time-domain
counterparts were considered only very recently, see [4,5].
This talk is dedicated to the derivation, stability and convergence analysis
of a time-domain Foldy-Lax model for sound-soft scattering by small obstacles. We start with the analysis of the time-domain counterpart of the respective frequency-domain model for circular scatterers from [6] and show that it is unstable for
some geometric configurations. To stabilize it, we propose its reinterpretation
as a perturbed Galerkin discretization of a single layer boundary integral equation.
Its unperturbed Galerkin discretization is then automatically stable due
to a coercivity-like property of the underlying operator and thus serves as a
basis to derive the stabilized model.
Let us remark that this reinterpretation provides us with an alternative way to derive asymptotic models as Galerkin discretizations of boundary integral formulations with well-chosen basis functions.
We will present the convergence analysis of the new model,
discuss its numerical implementation, and illustrate our findings with numerical experiments.
[1] P. Martin. Multiple scattering, vol. 107 of Encyclopedia of Mathematics and its Applications, Cambridge University Press, Cambridge, 2006.
[2] D.P. Challa, M. Sini. On the justification of the Foldy-Lax approximation for the acoustic scattering by small rigid bodies of arbitrary shapes, Multiscale Model. Simul., 12 ,
pp. 55–108, 2014.
[3] D.P. Challa, M. Sini. The Foldy-Lax approximation of the scattered waves by many small bodies for the Lamé system, Math. Nachr., 288, pp. 1834–1872, 2015.
[4] H. Barucq, J. Diaz, V. Mattesi, S. Tordeux. Asymptotic behavior of acoustic waves scattered by very small obstacles, ESAIM Math. Model. Numer. Anal., 55, pp. S705–S731, 2021.
[5] M. Sini, H. Wang, Q. Yao. Analysis of the acoustic waves reflected by a cluster of small holes in the time-domain and the equivalent mass density, Multiscale Model. Simul., 19, pp. 1083–1114, 2021.
[6] M. Cassier, C. Hazard. Multiple scattering of acoustic waves by small sound-soft obstacles in two dimensions: mathematical justification of the Foldy-Lax model, Wave Motion, 50, pp. 18–28, 2013.